?kvs.,xyech}".i.n,'g;T,e,c,h,' ,,. ,
On Quasi-Unitary Algebras with Semi•Finite Left Rings
By Eishl HoNGO
(Received Novcmber 1, 1956)
1. intrcductiek
In his investigations gf grottp ftlgebras en (not necessarily unirnodularÅr localiy corri- pnct topo]ogical groups, J• l)ixmier introdnced the concept of quasi-unitary algebrms, and studied their structure [i]. This nlgebras contain as special cases the unitary algebras, and the former notion is more generai in the point of view that eorresponding Ieft rings need not be semi-finite, i.e., it can possess a non-trivial purely infinitts component. In the studÅrr ofquasi-unitary nlgebras an important r6Ie is played by corresponding left rings, then the above fact gives a difliculty to us. Therefore it is reasonable t" confine ourselves to the case that the left ring is semi-finite. These quasi-unitary algebras admit adetaSlecl investigation by the geneyalizatigR eÅímaRy preperties ofunitary algebras, aRd L. PukaRszky deve!oped varicu$ theerems fr"m t}}is pgint ef vicw.
Recen;ly T. Oggs3wara has ebtained scme results cBRcerniRg the relatismship betwceR ll'-a}gebTag aftd comple{e quasi-unitary algebrms [3], Merg preciscly, he hasobtained thc fellg"'ing res"lts: (l) A complete qttasi-unltary algebr.ft A can be renormed in such a way t}]at lt becomes an ffes-algebra with the same inv•olution, and a positive definite operator AiERg (right ring ofA) is uniquely deterrnined in such a way thNt
Åq.x-"tÅr=(x,.aft}r),År, 1=:?s`fAf-i, sAfs :,g{i, s.nc=.xce,
Where Åq,År and (,År are re$pectixrcly ixxRer gereducts ef the qua$i-unitary and the llpitar3' algebva. (2År CDnverse}y, ifA is a= H"-algebra, Rxxd Mis a pesitive defu}ite "perater i= Rd, then A bec" Res a complete q=a$}-kkitary a}gebra ;\ith the iRncr prcd"ct akd the autcmgrph'"
ism1defi"ed by the kbgve eq"a!kies•
Tl}e "urpesc ef thls l}aper is to extend the re$ults of "jr. Ogasawara to the relationship betWite: itkitary algebras and quasi-unitary algebras with sitmi-finite left rings-
lft the second section, Nve shall list definitions of unitary and quasi-unitary algebras- and SOMe preliminary results which are needed later. In the third section we shall give the i rnaM reSults• The main results of this paper are as fo11ows: Let R be a quasi`unitary
algebra with a serni-finite !efe ring Rg (respectively uRitary algebraÅr, then there exi$ts a cie"Se quasi-unitary (resp. unitary) algebra Rl of ,SÅrR with the prgperty that Rl can be re'- nO rMed (not neces$arily equivareRtÅr ik such a way that it becgmes a gRitary (:eSP• 9Xa$i-
Sr'Phl)",g,g,e,P,:,x.,Pi,ii}Mes.{}E,,is?.g/r:n,s",e,,sg?:.cge,/j,lg.reke.2",i$g';g,ecte'.y.a.iEe,i:ll'lk.w;5tl.}
resUitS !Vil! be cb{aiked iii this sectlefi as a speciai case ofour resttlts obt"ined in the third
1
$ectiep. Thrgughout this paper we shal1 assume thd elementary results obtained by Dixmier and PukEn$zky.
2. Prelimingry couslderati"ps
An associative algebra R ever the cemplex R"mber field i$ caUed gww:uaitgry if there exist an involutive anti-automorphism x-xf, att automorphism x-tul and an inner prodact ( , ) such that R becomes a pre-Hilbert space sutisfying the foJlowing axioms:
(i) (xX xS) == Åqx,sc), (iO (x,s,)l}ll;e,
(iii) Åq.fry,.#i)-ny(f,:cs'i.--,År,
(iv) the left multiplScaticft pt.cy w!{h fixcd .x is coxxtipue"s, , -
(v) the lincar cembinatiens efthe elemeftts pf.the form cy+(izy)J3re de#se iR R (x,y,x arbitrary in R).
A unitary algebra A is a quasi-unitary aigebra with di =x. More preci$ely, an as"
sc}ciative 31gebra A over the complex number field is called une"tary if there exist an involutive antiautemorphisrn x- :S and an iRxer preduct ( , ) such that A becomes a pre
Hilbert space satisfying the fo11owing axioms: , -
( i )r (xS, xS) = (x,x), (ii)' (ry,=)=(y,x'x),
ÅqM)' the.left multiplication y-ay with fixed x is continuous,
Åqlv)' the elements of the form aEy are dense in A (x,pt,x arbitrary in A)•
'
If the Bnitary algebra A is completc, thexx (i)'--(iv)' are equivalent to (i)1 (ii)' and
(v)' xSx=e implies x=O foT scEA
Therefore A becemes an (prcper) H"-algebra ef Ambrese i= whice the inneT prodUCt has becri replaced by a suitable one in $uch a way that the multiplicative prePCrtY
l] ryll;l:L,illSi:ltllil:ll .hOql.dSdsi....itary algebra a=d let sft i}Åë the Hilbert $pace whicl; is ObtaiPed
by the cempledon ofR There exists a bounded operator U. (resp. Y.) on SR satSsfyiPg U'pt
.-' .`.ry
kigrtii,l:6k,l(aVtttiitXdil•.ier,e.",gXXii:G.IIFboT.h..ieft,E'6:?6s'l;:ig:.',zie.g",(ge,s.",',.R,2',eg,",fi'2tki
ef alt crperator's in the ferm U. Åqresp. V.). Rg and Rd are cemmutants "f each ether! MOce
]pmeciselY, the sec of al1 bounded oLperators in SR which cemmute with every elemeBt efR
On Quasi•Unitary Algebras with Semi-Finite Left Rings . 3
is Rs. The minimal closed extention Jof the correspondence x-xi is a positive, self-adjo•- int and non-singular operator• Denoting by S the continuation ofthe correspondence x..x'S over tbfl, then 1-'=SJS• An element a of ISi)R is called left bounded, ifthere exists a boun- ded operator VT. such that Uax= V.a for every xER.
A necessary and suMcient conclition for the semi-finiteness of the left ring Rg ofa quasi- unitaty algebra R is the representability of .I in the form [?lf'M-]], where ltf is pesitive, self-adjoint and non-singular operator belonging to Rd, and .nt'=SillS Åq[ ] denotes the mi- nimal closed extension provided it exists). Given a semi-finite eperator ring IN on a Hilbert space, a positive, self-adjoint and non-singular operaror H belenging to tsT, and a maximal normal trace cp defined on a two-sided ideal m (N, there exists such a quasi-unitary algebra R as to be i-isomorphic with N, and M' and the maximal extension ofthe canonical trace respectively correspond to Hand op under this isomorphism.
Ifn is a two-sided ideal in an operator ring, then nLl denotes the two-sided ideal formed by the elements Tin the ring for which 7'nTEn. Now let {p be the canonical trace for a semi-finite left ring Rg defined on the strongly closed two-sided ideal m in R-g where nt is thetotarity ofthe elements Y":'U.sUs, (ai,bi are left bounded and in D,v). We introduce for X,YEml the inner product (X,Y)={p(XYve), then m" becomes a unitary algebra with the involutive anti-automorphism .TX"-ÅrX'. We denote this algebra again by t:tJ-'.
3, Main results.
Let R be a quasi-unitary algebra with a semi-finite left ring Rh', then there exists a POSitive self-adjeint and non-singular operator IS/I belonging to Rd such that J--" [M'ISrl-i]
SvhereM'=SMS. Nowleti;xdEK (EC ERS) be the spectral representation of M'. The
imsii'1.8,t.;,1Åq;'i.ii1,t•"i1iiO,/g.=X=.eE[llrl.{'i---.r2111tiil?,{A",)l//t9/e.:.ilS:tsBal,i•Ibl'ijaClolP,R,!,,'1".w'r•//'l-oGf:ilAs=s
Ri an inner product, an involutive anti--automorphism and an automorphism as follows:
(X,]Yi) =op((XM',-:(A))(YA•r"(A))') for X,YER.t, iMS=M'-i(a)X"M'(ri) for XERn,
XJ=M'(n)XMt':(A) for XER.d,
M'here AI'"(A)--- jd )LndEK. with these definitions Ri becomes a quLrisi-unitary algebra such that the left ring Rf is "•-isomorphic with Rg [4]•
LEMMA 1. R, is tsomthtpltit zvith a dense euasi`-unttav, atgebTa in "bR•
PROOF- IfTERi, then there exists a left bounded element a ofGÅrRsuch that T= Ub,
a=d thereferc E(AÅra=:E'(A)a=a for suitable A, where E(A)=SE'(nt)S [4, Lemma 9ll. We define new the cerrespg=dence ef Ri i=tg SR by \e(UesÅr-nyww"a, and deRote by Rl the image "f
", under the mapping Yr. Rl is a subset of the left bbuftded elements ef diR. This cerres- pondence gives evidently an algebraic isornorphlsm between Ri and R{, more precisely,
it gives a one-to-one correspondence between Rnt and R{ such that ' Yn(NX+ge-Y)=AVr(X) + paVs(Y), "(YX)=aSp(X)ikÅqYÅr where X,YG R! and X,pa G C (complex number field). To prove that •"• defines an i$e lterphism betweea Ri and R{ it remains tc shew that for e"ery
X,YGR,, (I) i;n(X')xPe mak'(JÅrLO, (2) iStÅqXS) =Y"'(X), (3) (X,n="isÅqX), ile(YÅr). New we shall prove these equalities
Ad (l): IfX= U.ERA, then g is aR elemeRt in !he dpmain ofMand is left bounded, therefore frem Lemma 7a in [l] we have
'!lft(Xni) ='tPr`(M'(A)J\EM"-i(A)År- •gs(M(A)U.Adi-)(A))
-- 'iis'( Uw (A }2g -- i ptÅr.) = (Ul.j) = aJ ,
hence "(Xj)=Vni(X).
Ad (2): By the same reasen as Ad ÅqlÅr, for every X= U.ERA we have
kis(.XS) = yT (M' ma '(A) U.Mtt'(Ae)) = 3 s(UAy- i fA },s.M"(zt))
-- "( UAf ("o nft-: {A)At, {n)Af- : Åqd)sa) = xPH( Usut) "= Sa,
hence yf(XrS)="S(X).
Ad (3): Since ip(U.Ub'År=(Ma,Al5År [l], we have
(X, IYÅr = rp( Ua M' pt i (A))( V61Sf' - :(td)" )) = g)( UN pt i {d )a( Uft, ' : (n )m)b
== (M(A)M•- i (n),b nKA)ntt-- i (A)b) ta (a,b),
for every X =U., Y=VbffRi. So we obtain (X,Y)=(alff(",ft;e(XÅr). Thus ik is aniscmeT' phism betwecn the quasi-unitary algebra Ri artd the quasi-mitary algebra Rl in tbft• T"
cgncl=de the prcof of this !emma it remain$ only to sh"w that Rl is clense,in SR• SiVe define new by Yfia(;YÅr=:XM'-'(A)(Xff RD a gRhary mRpping •klel of R, in mi". It iseasxlY set:n that the mapping -tinl can be extended to a u#itary mapping a;fi, betweeR the space ORs
. and Sml. Therefore the image of Ri in tSint is dense in S}i:i}. Since the correspendeRCe Ma--)bU. can be extended to an isomorphisrn Yn2 between tha space ,SR and Otnl, the image ef•{pt(R" iR Sft' is dense in liÅrk OR the other hand, for X=U.ERA we have
a;"2Åqrxki(U")) = '"2( Ua M"- Z (A )) = '!Pn2( U"f- ! {A ).) -'-- tvI(A) M- i (ri )a = a,
therÅëfore "2"i and ",Jr coincide on Rt. And thus ft;r(R,)=Rl is dense in ,btt. This ceM' p!etes the proof of the lernma.
LEMMA 2t Ri con be renermed (net meesson"IJp e4ut'valens) g)} aR iRner pfedues in susk a tVa7
that ii ' 'bemas a tzttitacT algelirth
On Quasi•i.Jnitary Algcbras with Semi-Finite Lcft Rings 5
PRooIr. We intreduce eR RiahermitiaB bilinetar fufictional by tp( LrY`S). For X,YERi we have
tp ÅqXiY") == ip(XM, -i(AÅrYveAtltÅqA)) = cp(M,ÅqA)YIMi -T i(A)XX) = ,p(YMi - J ( ri).x"i" s,,fr(A)),
atld
,7)(xxrS)=ff}(XAte-:(A)XNAT(A)) =(7)(M,i(A)xne-":(ri)Ar-,T(A)xeefi•pt:i(A)) ,,. (p(rw l'(,d)xM, -' l"(A)(fif- "f(ri)xAf"p "-`(A))x)}2; o.
Fgrthermere, if cpt,(XLISLrS)=O thefi from the faithfulness ofqp it foliows that Afil`(A)XM'di(A)
==O, and then X'= O. Therefore op(XX$) is a$trlctly pg$itive syrr}met:ic bilinear functienal defined pR Ri. Ngw we $hall $hew that Ri becemcs a unitary algebra "rith the inner pro- dact defincd by ÅqXl',YÅr=rp(XYS), and with the involutive anti-•automorphism Xl---)keXS.
Ad (i)' in the definition of the unitary a!gebra. Fer every Xin Rri we have
Åqxs,xsÅr=op(xsx) =op(xxs)ttÅqx,xÅr.
Ad (ii)': For every X,Y,Z' in Rd, we have
xÅqX Y, ZÅr = rp(.Y YZS) = ep(YZSX) = op(Y(XSZ)S) = ÅqY, XSzÅr.
Ad (iii)f; Let X'=:Il., theA for eirery X,YER,t "re have lI .XYIi2== Åq. if IY7, Xr YÅr : {p(X YM -• 3(A) Y- XesM(A))
.,,, g,(A•f' - E• (A) yXM' "(A)M' -- "t(AÅrx Mg" l•(A)Aptt(A)x?lf- lÅqA)]lf' l`(.d) yAd' - "ÅqA)År : ,p (M, - J: (A) yac Mt e(,D (fi•f i l(A)xM, - li (ri)) ee (.M, a(A)x7M, -t l•(,d))M, l•(A) lyNY -- l(A)).
ifT and IZV .are bollRded eperaters i= 2tgÅrR, then for every elemeRt iR SÅrft we have
(TXTscTZTS'.:,.) =llrr.IF;SiTl2liIZS'ptil2 = l Tl 2(T' ac tT"x,x),
therefore O;;:SIT'- P,TT';:$lTI2T'NT', wherc l l is ordinary ope:atgr norm. Applying this faCt t" the abeve form, we have
ll XY" Sg$1 l M' "l'(A).YM' " "(A) l -2 si ,( YA•I"' i( A)paM`(A)) = l Unvi (d )nt "" "("t )a l 2N YIi 2' = l UJ ba l 2 il Yli2= kIl Yll 2,
Whe':lkdÅq;,g)e,\,ds.e,ngy,s,iX,ifl6"2.`:e,:,}s,p,pg",g,ll:,;,il,IY..iSiR",,"',i.".9]lj,l",r.fi.X:,dhodX'.d.pt,d
bY Godement iR [2]. IL,et XERrt, then we have , h.
rp(XXB -- ge(xM'- i(.tDxNM'(A)) = q(M - e(A)x"M'l(A)M'i(A)LxM' p lÅqA)År
"r${ l M' ]:(A) l 2ip(Af"- ti (A)xxxtlir '` 4'(d))= l .itl'lÅqd) l :qj(xM'- lÅqA)M'- li(A)xee) ;;:$ l .ntl'"(AÅr l 2 l M'- E-t (A) l :Åqp(xxua ).
Now let X be an eiement in Ki and let X= MH be the canonical deccmpesit!cn of v7iC.
Then H=(XXX)l=:I:MEA and Mis partially isometrie. It ls clear that rr(A)H=HE'ÅqAÅr
==Hand E' (A)M•--- JJ7e(A) =X; and therefore ff,IPERi. Define JI. tand X. as fo1!ews:
Hn=Iill..,ÅrldEK, Xn= MÅqHJ + ""'' ÅÄHlnÅr•
TheR Hn and Xfi are elements of RA and from the abeve assertien we have ÅqX -- Xn,X- XnÅr =' cps(ÅqX' -"- Xn)(X- Xn)S) = {p((X pt Xt#)M`-i(A)(.IY -- Xg)M'Åqd))
;Sl l M' l(A) l 2 l M' -` a (ri) l 2p(ÅqX- "--- X.)(X - Xn) ee),
svhere A is an Snterval determined cnly by X7, Frem rp(ÅqX -k Xn)(X - Xn)") =' IXW(Hi) s
pÅrn we have
ÅqX '-' XrmX- X.År$ l .M'"(AÅr I2l Mt-l(A) i 2=rp(,H",?).
pÅrn
Fyem the fact that the interval A is determined enly by X, M'lÅqA) and M'ny l(A) depend gnly en Xand Ret depenc! "p n. The sequence Xgp(HLI?) is convergent in 2b;n!, se we obtaiR
pÅrn ÅqX-Xn,X'X.ÅrÅqG for suMciently large n. 'I7hen the set ofthe elemeatsefthe fcme XY where X, V in K is den$e Sn Ri with the Berm defiRed by the inner prod"ct Åq , År. This cornpletes the proofofthe lemma. We denete this unitary algebra by R2i
Su!nming up the above lemmas, it .can be seen that ifR is a quasi-"nitary algebTa with a semi-finlte left ring Rg then there exists a dense subset Ra in ijR such that it becpmeS 'a unitary algebTa wkh the samtt involutiQn by the inner product defined in iemma 2- . ! ow we intend te give mere precise iRformation about the connection between the
inner preductÅq,)ef Ri andÅq,Åref R2. For ever9 X,YERA we have '
(X,Y)=ip(Xllf'-](A)(YM'-}(A))es)=op(XVtlS-2(A)YX) . =,p(xMi-i(A)M'"i(A)YftM'-}(A)M'(ri)) L
=gp(Xlltf'-3(,ett)(M"}(A)YMr'-3(A))"M2(A)) ,
=tp(X(M'.iÅqA)YM'-}(2DÅrS)=:Åqx,M"ti(A)YM'-2(A)År. v i
We dm(xc by T thc minimal c}osed extensien ef the operater VM,-i 6ft :ii}R,, "then from the
faitt that YrM,-i !ir =: Upt"iÅqA)x UM-bk(ti}. this eperater cerresp6ndes to the epeyafer M"i bn
di'x;"rmtkf'fithe imctrphim -hgp.; Ifw"e' put T'xS7TS, then we heve " li ' '
On Quasi-Univary Algebras with Serni•Finite Lcrt Rings 7
m"-xr `ke- y,,,- i (., }s lzw - i {. }s.\= yw - i {,t )s yys,, -- ; {d)nttS - iÅqn).x es ltff(zf)
,,. Mt- ]ÅqAÅr.\Mt-]ÅqA) = UAf- !Rf.-- :.-
New "'e have
(ct,b)=(Iila,Ub) =ÅqUa,M'-"}(A)UbM'-:(A)År"=Åqa,Bl-i(AÅr21tl'-]ÅqAIÅrbÅr
==tr Åqa,Mmu' iA,I' nt'bÅr. ,
This equality combined with Lemma 1 and Lemma 2 gives the fo11gw!ng thegrem.
THEoREM i. Lct R be a auast'-tmitap, algebra with a semi-Yinite ioft ring Rg, then there e,xists in ,g.)R a dcruse euai-ecnitapp algebra Rl which can bc renoi7ned (not riecessatil;3i ege{!'ralent) in such a estg;y that it begames a unitarv algebra R5 zvith tke sa#ie ine'elugien. Furtliuanere, ;t2e dcRete b], ( , ) aRdÅq , År i;z;2er prgd:ets gf Ra RRd Rl. resPecgivel],f Shca there fxisSs a Pest"Sive, sgif-Edy'einS end non-stngfilar operater M' bdgRgiRg te the k.ft rS.rrg R" sivak Shat
Åqa,b) =: Åqa,M fi iM"-'bÅr, J = [M'M"!],
l.Xf"w wc $hftll c"R$ider the ccftvcrse t" the abcve the"rem. Let.R be a uftitary algcbr3, then by the thegrcm 3 iR [2] .there exi$t$ " Rgrmal, faithful llnd maximal trace pe "n tl}e left ring Rg sllch that
rv( U. U# )m (a,b)
for every bounded element in Ofi. By the trace {p aRd a positive, sclf-aajQint uRd poR- singular operator M' belenging t" RC we ccmpesc the quasi-unitrary algebTa R2 defiRed at the beginging gf the present sectlett, lfXE RA, theR theye cxists k projectigtt E'(A) $ueh th3t
rr(A)Xr== XE'(A)=X, therefore XXrr(A)=e(A)XX= X" and then X"EBA. Thi$ fact
SboWs tl;at the set Ri is cleseci "nder the eyseration ee. New we Åëen$ldcrr the sct R"vith alge- braic "nd topological structures ofmi-`, then fti becemes a unitary algebra with the irmv"lu"
tiVe anti--automorphism X.X". We clermote this unitatry algt)bra again by Ri. It is easily seeMhat the ene-to-one mapping U.---a ef Ri in the bo"nded unitary algebra ef R (the totality cfthe bgunded elemeRts ef R) preserve$ the algebrRie eperati"Rs axxd the iRv"lutive aRti"autgmerphism. Tha image cfRt wndey thi$ mappl=g is dexgted by RS• TheR RI is agnitary algebra with the ifivelutive afiti-automcrphi$m dcfincd•by the restrictioR ofthe
CgXinuation S of originai involution to Rl. We intrcduce in Ra axx Snner produet by
Åqa,bÅr :Åqa,M-iAfmo}b) • L
Where M tc SMts, aftd clefine an "perater x by thc rai=iraal cl"sed extensi"n "fthe restr.icti,en 2ge"ilfau"tXmigeSiZdmÅr":.S3at,t',P::e,i."lli`.h,?•.Wti."'lhlllttiC.ll•l,es.,:'gy.'/igX"k?SYFitlli{ithbS'gk"•:chi\'a`',i
cheW first thatJis an automorphism en RS. rf U.,VbGR,t foti an ipterval A,'thCR
J(t b) =M(A)Mkj(A)ab= M'(d)M'"i(A)U.b=M'(A)UutM-i(d)b
•"s'- UMx (AÅrAf-i{fiÅr.M'(dÅrM"" ](A)b = : (M'(A)M'2(a)a)(M'(ad)M-i (A)b)
==(Xa)(Jb), On the other hand
Zlla = UAf-s{A}M,-'ÅqA}e=M'-i(A) VaM'-l(A) E Ri,
theTeforelis an automerphisrr} in Rl. New we preceed to the preofof the fact that Rl is a guasi-unitary algebra.
Ad (i) in the definititm ofthe quasi-unitary algebra. Let U.eR,f, then we hxxve
Åq2,,S,.SÅr = (,,,S,M- i(A)Mi -:(A),ifr) = (S., SM' -` }(A)M '"- i(AÅrx)
=(M'-':(A).Mei(A)x,pa)=(x,M'-'i(A)M'-2(ri)x)=Åqx,rvÅr, hence we obtain Ad ÅqiÅr,
Ad (ii):
Åqx,xjÅr = (x,M':(AÅrtllf'-](A)M'ÅqA)M'i(A)xÅr -- (M -" i(A ).x,M "' i(A )-x)). O
where U.ERa for an intervai A, Acl (iii): If U.,U.,l7.GRd then wct huve
Åqay,xÅr = Åq,ry,M-k(A)Mf -' i(A)-: ) = (x U.M- M-i(A)Mr ny }(n):) = (x MF"(zt) USM' -i(A)aÅr = (y,A•f -i(A)M"- i(A) Ubfi (A ) Af- i (A Årsa:) = (y,M"}(A)M' "` :(A)(JSx)x) = : ÅqÅrr,x'f:År.
Ad (iv): Fer l7.,VrERA we have
Ilayll2==Åqay,ttiyÅr=:(xr,M-i(A)M'-i(AÅrcy)=(UsUy,UAi-i(d)Mr'"fenx7)
=:(U.U,,M'-i(A)U.U.M"'i(A))={sp(U.U.Mi"iÅqA)U;JrUIM"2(A)) ,
-- q)(M' -" l(AÅrv#Mr -- l(A)Mi l(A) v.-Mr - l(A)Mx - iÅqA) u.Mr i(A)Mi -- "(A) u,nf' -4(rt)) , = I M"'-?XÅqA) Upat l(A) i 2,p(M i -- i(A) U#MS •-- 3(A) U,)
-- l UNI(li)ntt-"i(d}.I 2Åqy,crÅr= l Ux-g. Ii L'Åqy,crÅr =k2llyil L',
where k depimds only on x.
Ad (v): Since1is the minimal Åël"sed extension of the product of twQ positive, sel.f"
adjeint andi nep-singular operators M'} and M', J is itself positive, seif-adjoint aRd POn'
giugu!ar. So•'the rartge ef the eperator l+J is SR. To prove Ad (v) it is suMcient tDShOW
On Quasi-Vnitary Aigebras with Scmi-FiniteLert Rings 9
thatJ is the minimai closed extension of its restTiction t" the set formed by the lincar cem- binations of the eiements of the form xy where tv andy are,arbitrary elemeftts ifi R'2. Let X=U. EE RA, then from the fact that ES(A) is the leRst upper beufid of projectlon in m'}
there exists a projection P:SE'(A) in u"' such that P.X"= Up.E Rn is arbitrary mear to X in the metric of tSsi:". For P= Up Ei R.f "'e hasre
-1"X:x -.,XR.v!l L' = Åq1( .x -p.xÅr,J(x - p.xÅrÅr
= (M'(A)M "- )(AÅr(x -- p.x),M-i(A)M' "' iÅqAÅrMr( A)M-i(AÅr(.-p.År) -ny (M'(A) Ui epuxM' "- i(A),Al '`'Åq K{ ) Ux -px)
= cp,(M'( A) U. .-,.M'- ](A)Ui -..M "' 2(A))
=: (p(jf' l' (A)A;f -" '(A)U. .- ,.Al'- :'` (A)M'-"(A)IJ#' -i,=M-'(AÅrM "- }" (stl )År
ves l M'- ti'`(.,d) l L' l M"" '(A)M' }(/t) 1 :' {1:,( U. -. p. U.X.,,.)
= ir.(A)L'k'(A)L'q,((X'- PX)(X-- PX)XÅr.
$ince PiY EE R.i, AT(A) and'Js.t(A) depend gnly gR X. TheTefere there exist$ a prejectiei} Ps"ch that lp.v ig aybStrza"y Rear te j?1.x in the metric oÅíthe ]Er!ilbert space whick Ss ebtained by the cgmpletioR cf R:. Tl}en by the defini{ici} cfl it is clear that1is the mifiimal ciesed ex- tensieR cf its re$trictifiR te the linear set in tSR,. formed by t};e Milear combination of the eiements of the form .vy. 1"hen the foilo"ring iheerem holds
,
1"HEOilEM 2. Let R be a unitaq, atgebra and iet n•i' be aPositie'e, self-adj'oint and non-singular
t
Optrator beiongirrg to the left ring Rg, titen there e.vists i?t agbg a dense !mitarv algebra Ri such tiutt this sc'aigebra becomes a esttesi-unitap, algebra zvitli l?2s innerProduce
Åqx,ptÅr = Åqa-,fyfm:M'-ipt) (M== sM'sÅr
and the ctiitomerphtsm 1 defined b;y the minimal ciosed extemsion of tire restn'ctien of nf'fif"i in Rl.
N?ow we remark that the structure qf the quasi-unitary algebra ebtained by thÅëabeve Method is closely re!ated te the n3ture of the eperater M'•
4- Complete quasl-tinitary algebras-
het R and R' be two quasi-unitary algebras. We cnii R is a continuntion of RI, ifR' iS a dense subalgebra ef R, and also if the inner product, the automorphism and anti'aUt?
MerPhism in R' are the restriction ofthe corresponding notions in R• R is said to be Maii"
Mal' ifit h`rLs no preper centinuatigR. R i$ max. imal ifand only ifit pessesses the followjn.g PrfiPertY: lf for a EEI ,Sft, J"a exists aRd is !eft bounded for every non-zere integar, then a iS
ii•::'illlll.;':"ibee,l/:i,m,,';xikl\ihi,2".zsahy.":,t",rx,al..ggb;,i•,megt.fg's",.',rs,1'?m.'n"i--",(Yll}`.".'".IK,'d,'g
dense subalgebra efR. Ncw we suppese that the operater M' treated in the above section is bottnded, then we have
Ilxji 2 -- Åqat,ptÅr = Åq U., U.År =: rp( U. U.S) --- rp( U.M' -' i V#M') ;:iS l M' E l 2 l M' l I 2{p( U..A,f` '- 2 U.ee)= l M' } l `ep( U.M "- i( U.M'-i)f)
-ny"' k`( U., V.) -'ny• '--- k`(x,.r) = h`Uixiil 2,
